Problem of the Week

Updated at Jun 29, 2015 9:15 AM

This week's problem comes from the calculus category.

How can we find the derivative of csc⁡xcos⁡x\frac{\csc{x}}{\cos{x}}?

Let's begin!



ddxcsc⁡xcos⁡x\frac{d}{dx} \frac{\csc{x}}{\cos{x}}

1
Use Quotient Rule to find the derivative of csc⁡xcos⁡x\frac{\csc{x}}{\cos{x}}. The quotient rule states that (fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}.
cos⁡x(ddxcsc⁡x)−csc⁡x(ddxcos⁡x)cos⁡2x\frac{\cos{x}(\frac{d}{dx} \csc{x})-\csc{x}(\frac{d}{dx} \cos{x})}{\cos^{2}x}

2
Use Trigonometric Differentiation: the derivative of csc⁡x\csc{x} is −csc⁡xcot⁡x-\csc{x}\cot{x}.
−cos⁡xcsc⁡xcot⁡x−csc⁡x(ddxcos⁡x)cos⁡2x\frac{-\cos{x}\csc{x}\cot{x}-\csc{x}(\frac{d}{dx} \cos{x})}{\cos^{2}x}

3
Use Trigonometric Differentiation: the derivative of cos⁡x\cos{x} is −sin⁡x-\sin{x}.
−cos⁡xcsc⁡xcot⁡x+csc⁡xsin⁡xcos⁡2x\frac{-\cos{x}\csc{x}\cot{x}+\csc{x}\sin{x}}{\cos^{2}x}

Done
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